Poisson-generalized geometry and R-flux

We study a new kind of Courant algebroid on Poisson manifolds, which is a variant of the generalized tangent bundle in the sense that the roles of tangent and the cotangent bundle are exchanged. Its symmetry is a semidirect product of β-diffeomorphisms and β-transformations. It is a starting point of an alternative version of the generalized geometry based on the cotangent bundle, such as Dirac structures and generalized Riemannian structures. In particular, R-fluxes are formulated as a twisting of this Courant algebroid by a local β-transformations, in the same way as H-fluxes are the twist of the generalized tangent bundle. It is a three-vector classified by Poisson three-cohomology and it appears in a twisted bracket and in an exact sequence.